With one caveat, I say absolutely!
The caveat is...make sure he understands why people will be saying HB to him so many days after his b'day. My son wont' even wear his badge the day after his b'day...then again, we've actually been at the parks on his 4th and 5th b'days, so he's had a full day of well wishes, and by the next day he's just DONE with it.
So if your guy knows that people will be saying it to him, and he's cool with it...then absolutely go for it. After all, you'll still be celebrating it while you are at the parks; why not tell people?
And the "aside" to the caveat could be...you could get an "I'm Celebrating" button, to sort of say "we're celebrating that it was his birthday" (not that it will state that), if he might be uncomfortable with the Happy Birthday.
Oh! And also make sure they know it's his FIRST visit, so he can get the First Visit button!